{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Notes for Algebra with Python [WIP]\n",
    "\n",
    "The following are a set of notes for understanding some foundational concepts of Algebra using Python with a focus on ML. \n",
    "\n",
    "References: \n",
    "\n",
    "- [Chapter 4 - Functions and Algebra with Python](https://learning.oreilly.com/library/view/the-statistics-and/9781800209763/B15968_04_Final_RK.xhtml#_idParaDest-95)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions\n",
    "\n",
    "Functions are used to map from one mathematical object to another. In ML, it's important to understand the concept of function as we use them a lot. In fact, for a lot of ML concepts we are essentially tying functions together mapping inputs to outputs. Let's start with some basic concept of functions, then gradually make our way into the more common functions applied in ML."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example of a squaring function:\n",
    "\n",
    "$$\n",
    "f(x) = x^2\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# import your main libraries\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n"
     ]
    }
   ],
   "source": [
    "# simple squaring function\n",
    "def f(x):\n",
    "    return x**2\n",
    "\n",
    "# test the function\n",
    "x = 2\n",
    "print(f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's visualize what that looks like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fd7786b9d50>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# visualize using matplotlib\n",
    "\n",
    "x = np.linspace(-10, 10, 100)\n",
    "y = f(x)\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A constant function:\n",
    "\n",
    "$$\n",
    "f(x) = c\n",
    "$$\n",
    "\n",
    "with $c$ being a constant."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fd7811b9910>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# a constant function\n",
    "def f(x):\n",
    "    return 2 # constant\n",
    "\n",
    "# test the function\n",
    "x = 2\n",
    "\n",
    "# visualize using matplotlib\n",
    "x = np.linspace(-10, 10, 100)\n",
    "y = [f(x_i) for x_i in x]\n",
    "\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A linear function:\n",
    "\n",
    "$$\n",
    "f(x) = mx + c\n",
    "$$\n",
    "\n",
    ", with $m$ being the slope and $c$ being the y-intercept -- both constants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fd7e8539850>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# a linear function\n",
    "def f(x, m, c):\n",
    "    return m*x + c\n",
    "\n",
    "# test the function\n",
    "x = 2\n",
    "m = 3\n",
    "c = 4\n",
    "\n",
    "# visualize using matplotlib\n",
    "x = np.linspace(-10, 10, 100)\n",
    "y = [f(x_i, m, c) for x_i in x]\n",
    "\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Polynomial functions:\n",
    "\n",
    "$$\n",
    "f(x) = a_0 + a_1x + a_2x^2 + \\cdots + a_nx^n\n",
    "$$\n",
    "\n",
    "with $a_0, a_1, a_2, ..., a_n$ being constants. And $n$ is the degree of the polynomial."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fd768081750>]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# a polynomial function\n",
    "def f(x, a, b, c):\n",
    "    # a, b, c are coefficients\n",
    "    return a*x**2 + b*x + c\n",
    "\n",
    "# test the function\n",
    "x = 2\n",
    "a = 8\n",
    "b = 1\n",
    "c = 8\n",
    "\n",
    "# visualize using matplotlib\n",
    "x = np.linspace(-10, 10, 100)\n",
    "y = [f(x_i, a, b, c) for x_i in x]\n",
    "\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Logarithmic functions:\n",
    "\n",
    "$$\n",
    "f(x) = c\\log_a x\n",
    "$$\n",
    "\n",
    "with $c$ and $a$ being constants, $log_a$ is the logarithm function with base $a$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fd7d8cb16d0>]"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# a logarithmic function\n",
    "def f(x, a):\n",
    "    return a*np.log(x)  # natural log\n",
    "\n",
    "# test the function\n",
    "x = 2\n",
    "a = 3\n",
    "\n",
    "# visualize using matplotlib\n",
    "x = np.linspace(0.1, 10, 100)\n",
    "y = [f(x_i, a) for x_i in x]\n",
    "\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exponential functions:\n",
    "\n",
    "$$\n",
    "f(x) = c\\cdot a^x\n",
    "$$\n",
    "\n",
    "with $c$ and $a$ being constants."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fd7d8e30b90>]"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# an exponential function\n",
    "def f(x, a, c):\n",
    "    return c*np.exp(a*x)\n",
    "\n",
    "# test the function\n",
    "x = 2\n",
    "a = 3\n",
    "c = 4\n",
    "\n",
    "# visualize using matplotlib\n",
    "x = np.linspace(-10, 10, 100)\n",
    "y = [f(x_i, a, c) for x_i in x]\n",
    "\n",
    "plt.plot(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Function Roots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check if a specific function has root. Below we are checking linear function which has a unique root of $x = -c/m$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(-1.3333333333333333, 0.0, 'x = -c/m-1.3333333333333333')"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# function to find root of a linear function\n",
    "\n",
    "def f(x, m, c):\n",
    "    return m*x + c\n",
    "\n",
    "def has_root(x, m, c):\n",
    "    return x == -c/m\n",
    "\n",
    "# generate range of values that includes root of linear function\n",
    "x = np.linspace(-10, 10, 100)\n",
    "\n",
    "# add root to the list\n",
    "x = np.append(x, -c/m)\n",
    "\n",
    "# order the list of values\n",
    "x = np.sort(x)\n",
    "\n",
    "y = [has_root(x_i, m, c) for x_i in x]\n",
    "\n",
    "# y to visualize\n",
    "y = [f(x_i, m, c) for x_i in x]\n",
    "\n",
    "# visualize using matplotlib, with red dots at roots\n",
    "plt.plot(-c/m, f(-c/m, m, c), 'r.')\n",
    "plt.plot(x, y)\n",
    "\n",
    "# label root value with text x = -c/m\n",
    "plt.text(-c/m, f(-c/m, m, c), 'x = -c/m'+str(-c/m))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transformations of Functions\n",
    "\n",
    "Transformations is an important concept where you take the output of one function and put it through another function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Shifts happen when you move the function up or down, left or right. See example below of a vertical shift:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6fe80d0d90>]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# function f(x) = x**2\n",
    "def f(x):\n",
    "    return x**2\n",
    "\n",
    "# test the function\n",
    "x = np.linspace(-4, 4, 100)\n",
    "\n",
    "# visualize using matplotlib\n",
    "plt.plot(x, f(x))\n",
    "\n",
    "# plot function shifted by 1 unit vertically; use red\n",
    "plt.plot(x, f(x)+1, 'r')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can do a horizontal shift:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f3403b5d2e0>"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot function shifted by 2 units horizontally; use green\n",
    "x = np.linspace(-5, 5, 100)\n",
    "plt.plot(x, x**2, label='$f(x)=x^2$')\n",
    "plt.plot(x, (x-2)**2, label='$f(x)=(x-2)^2$')\n",
    "\n",
    "plt.xlim(-6, 6)\n",
    "plt.ylim(-2, 4)\n",
    "\n",
    "plt.plot(x, x*0, 'k', alpha=0.2)\n",
    "plt.plot(x*0, x, 'k', alpha=0.2)\n",
    "plt.legend()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scaling transformation example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f6fe61e0430>]"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# vertical scaling\n",
    "x = np.linspace(-6, 6, 100)\n",
    "plt.plot(x, x**2)\n",
    "plt.plot(x, (x**2)/2, 'g')\n",
    "\n",
    "plt.xlim(-6, 6)\n",
    "plt.ylim(-2, 4)\n",
    "\n",
    "plt.plot(x, x*0, 'k', alpha=0.2)\n",
    "plt.plot(x*0, x, 'k', alpha=0.2)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Scaled by negative constant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f340844c1c0>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# horizontal scaling\n",
    "x = np.linspace(-6, 6, 100)\n",
    "plt.plot(x, x**2)\n",
    "plt.plot(x, -(x**2)/2, 'g')\n",
    "\n",
    "plt.xlim(-6, 6)\n",
    "plt.ylim(-4, 4)\n",
    "\n",
    "plt.plot(x, x*0, 'k', alpha=0.2)\n",
    "plt.plot(x*0, x, 'k', alpha=0.2)\n",
    "\n",
    "plt.plot(x, x**2, label='$f(x)=x^2$')\n",
    "plt.plot(x, -(x**2)/2, 'g', label='$f(x)=-\\\\frac{1}{2}x^2$')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plotting sine function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f340823be80>"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-6, 6, 100)\n",
    "plt.plot(x, np.sin(x), label='$f(x)=sin(x)$')\n",
    "\n",
    "plt.plot(x, x*0, 'k', alpha=0.2)\n",
    "plt.plot(x*0, x, 'k', alpha=0.2)\n",
    "\n",
    "plt.legend()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another example of shifting transformation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f3403557160>"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-6, 6, 100)\n",
    "plt.plot(x, x**3 - x, label='$f(x)=x^3-x$')\n",
    "\n",
    "# horizontal shift\n",
    "plt.plot(x, (x+2)**3 - (x+2), 'g', label='$f(x)=(x+2)^3-(x+2)$')\n",
    "\n",
    "plt.plot(x, x*0, 'k', alpha=0.2)\n",
    "plt.plot(x*0, x, 'k', alpha=0.2)\n",
    "\n",
    "plt.ylim(-5, 5)\n",
    "plt.legend()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vertical scaling of sine function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x7f3403548490>"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-6, 6, 100)\n",
    "\n",
    "# sine with vertical scaling\n",
    "plt.plot(x, np.sin(x), label='$f(x)=sin(x)$')\n",
    "plt.plot(x, 2*np.sin(x), 'g', label='$f(x)=2sin(x)$')\n",
    "\n",
    "plt.plot(x, x*0, 'k', alpha=0.2)\n",
    "plt.plot(x*0, x, 'k', alpha=0.2)\n",
    "\n",
    "plt.ylim(-5, 5)\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises:\n",
    "\n",
    "- Plot the following function: $f(x) = x^2 + 2x + 1$ and find the roots (if any) of the function.\n",
    "- A good way to understand function is to understand more closely their properties. Play around with the function by changing values and observe how they behave. Are any of the functions related to each other? How? \n",
    "- Write a code that applies a [sigmoid](https://en.wikipedia.org/wiki/Sigmoid_function) function to a given input."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3.8.15 ('math')",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.15"
  },
  "orig_nbformat": 4,
  "vscode": {
   "interpreter": {
    "hash": "885f0b8c324fe4d130bb8744e2598b34a480bb115953c09edacbc3cda2096502"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
